As it took far too much Googling to find this, if you need to access the AppData folder for the System account, go here:

C:\Windows\System32\config\systemprofile\AppData\Local
C:\Windows\SysWOW64\config\systemprofile\AppData\Local
I hit this because we needed to clear the NuGet package cache for a TeamCity build agent which was running as a service under the System account.